Planar metallic nanoparticles separated by nanoscale distances from a metal film support unique plasmonic resonances useful for controlling a wide range of photodynamic processes. The fundamental resonance of a film-coupled planar nanoparticle arises from a transmission-line mode localized between nanoparticle and film, whose properties can be roughly approximated by closed form expressions similar to those used in patch antenna theory. The insight provided by the analytical expressions, and the potential of achieving similar closed-form expressions for a range of plasmonic phenomenon such as spasing, fluorescence enhancement, and perfect absorbers, motivates a more detailed study of the film-coupled patch. Here, we present an expanded analytical analysis of the plasmonic patch geometry, applying an eigenmode expansion method to arrive at a more accurate description of the field distribution underneath a film-coupled plasmonic nanocube. The fields corresponding to the inhomogeneous Maxwell's equations are expanded in a set of lossless waveguide eigenmodes. Radiation damping and Ohmic losses are then perturbatively taken into account by considering an equivalent surface impedance. We find that radiative loss couples the lossless eigenmodes, leading to discernible features in the scattering spectra of the nanocubes. The method presented can be further applied to the case of point source excitations, in which accounting for all potential eigenmodes becomes essential.